 Can any two matrices be added together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.


No, not any two matrices can be added together. For two matrices to be added, they must have the same dimensions, meaning they must have the same number of rows and the same number of columns. 

### Explanation:

Matrix addition is defined as the element-wise addition of two matrices. This means that for each corresponding element in the two matrices, the sum is computed. For this operation to be valid, every element in one matrix must have a corresponding element in the other matrix. Therefore, the two matrices must be of the same size.

### Example of Two Matrices That Cannot Be Added:

Consider the following matrices:

\[
A = \begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
\]

\[
B = \begin{bmatrix}
5 & 6 & 7 \\
8 & 9 & 10
\end{bmatrix}
\]

Matrix \(A\) is a \(2 \times 2\) matrix (2 rows and 2 columns), while matrix \(B\) is a \(2 \times 3\) matrix (2 rows and 3 columns). Since the number of columns in \(A\) and \(B\) are not the same, we cannot add these two matrices together.

### Summary:

Two matrices can only be added if they have the same dimensions (i.e., the same number of rows and columns). If the matrices do not have the same dimensions, matrix addition is not defined.

Would you like more details or have any questions?

### Related Questions:

1. What are the rules for matrix multiplication?
2. Can you subtract two matrices with different dimensions?
3. How do you perform scalar multiplication on a matrix?
4. What is a zero matrix, and how does it relate to matrix addition?
5. How is matrix addition used in practical applications?

### Tip:

When adding matrices, always ensure they have the same dimensions by checking both their row and column counts.

Explore whether any two matrices can be added together in linear algebra. Learn the conditions under which matrix addition is defined and discover an example of matrices that cannot be added. Suitable for students studying linear algebra or matrix operations.

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